LMFDB - Embedded newform 9065.2.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9065,2,Mod(1,9065)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9065 = 5 \cdot 7^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9065.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-1,2,5,-2,0,0,-2,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.3843894323$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.126032.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} + 6x - 2 $ x^5 - 6x^3 + 6x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1295)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.392048$ of defining polynomial
Character	$\chi$	$=$	9065.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.392048 q^{2} -2.20143 q^{3} -1.84630 q^{4} +1.00000 q^{5} +0.863067 q^{6} +1.50793 q^{8} +1.84630 q^{9} -0.392048 q^{10} +0.255115 q^{11} +4.06450 q^{12} +1.33836 q^{13} -2.20143 q^{15} +3.10141 q^{16} -0.276162 q^{17} -0.723838 q^{18} +5.26593 q^{19} -1.84630 q^{20} -0.100017 q^{22} +0.173848 q^{23} -3.31961 q^{24} +1.00000 q^{25} -0.524703 q^{26} +2.53979 q^{27} -10.2196 q^{29} +0.863067 q^{30} -2.40286 q^{31} -4.23177 q^{32} -0.561619 q^{33} +0.108269 q^{34} -3.40882 q^{36} -1.00000 q^{37} -2.06450 q^{38} -2.94632 q^{39} +1.50793 q^{40} -1.01587 q^{41} -6.34995 q^{43} -0.471019 q^{44} +1.84630 q^{45} -0.0681570 q^{46} +1.32389 q^{47} -6.82755 q^{48} -0.392048 q^{50} +0.607952 q^{51} -2.47102 q^{52} -5.58325 q^{53} -0.995722 q^{54} +0.255115 q^{55} -11.5926 q^{57} +4.00657 q^{58} +6.83412 q^{59} +4.06450 q^{60} -1.59348 q^{61} +0.942038 q^{62} -4.54377 q^{64} +1.33836 q^{65} +0.220182 q^{66} -4.43838 q^{67} +0.509878 q^{68} -0.382715 q^{69} +1.45974 q^{71} +2.78410 q^{72} +6.54377 q^{73} +0.392048 q^{74} -2.20143 q^{75} -9.72248 q^{76} +1.15510 q^{78} +2.54403 q^{79} +3.10141 q^{80} -11.1301 q^{81} +0.398269 q^{82} +3.29433 q^{83} -0.276162 q^{85} +2.48949 q^{86} +22.4977 q^{87} +0.384697 q^{88} +10.1693 q^{89} -0.723838 q^{90} -0.320976 q^{92} +5.28973 q^{93} -0.519029 q^{94} +5.26593 q^{95} +9.31595 q^{96} +9.02672 q^{97} +0.471019 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 2 q^{4} + 5 q^{5} - 2 q^{6} - 2 q^{9} - 7 q^{11} + 4 q^{12} + 3 q^{13} - q^{15} - 4 q^{16} - 5 q^{17} + 2 q^{20} - 10 q^{22} + 4 q^{23} + 8 q^{24} + 5 q^{25} - q^{27} - 17 q^{29} - 2 q^{30}+ \cdots - 2 q^{99}+O(q^{100}) $ 5 * q - q^3 + 2 * q^4 + 5 * q^5 - 2 * q^6 - 2 * q^9 - 7 * q^11 + 4 * q^12 + 3 * q^13 - q^15 - 4 * q^16 - 5 * q^17 + 2 * q^20 - 10 * q^22 + 4 * q^23 + 8 * q^24 + 5 * q^25 - q^27 - 17 * q^29 - 2 * q^30 + 8 * q^31 - 10 * q^32 + 3 * q^33 + 24 * q^34 - 20 * q^36 - 5 * q^37 + 6 * q^38 - 13 * q^39 + 10 * q^41 - 6 * q^43 + 2 * q^44 - 2 * q^45 - 16 * q^46 - 11 * q^47 - 2 * q^48 + 5 * q^51 - 8 * q^52 + 14 * q^53 + 2 * q^54 - 7 * q^55 - 26 * q^57 + 18 * q^58 + 4 * q^60 + 4 * q^61 - 4 * q^62 - 16 * q^64 + 3 * q^65 + 12 * q^66 - 28 * q^67 - 12 * q^68 + 10 * q^69 + 10 * q^72 + 26 * q^73 - q^75 - 4 * q^76 - 12 * q^78 - 17 * q^79 - 4 * q^80 - 19 * q^81 - 4 * q^83 - 5 * q^85 - 16 * q^86 + 5 * q^87 + 6 * q^88 + 2 * q^89 - 18 * q^92 + 24 * q^93 - 22 * q^94 - 10 * q^96 + 35 * q^97 - 2 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.392048	−0.277220	−0.138610	−	0.990347i	$-0.544263\pi$
$2$	−0.392048	−0.277220	−0.138610	+	0.990347i	$0.544263\pi$
$3$	−2.20143	−1.27100	−0.635498	−	0.772102i	$-0.719205\pi$
$3$	−2.20143	−1.27100	−0.635498	+	0.772102i	$0.719205\pi$
$4$	−1.84630	−0.923149
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.863067	0.352346
$7$	0	0
$7$	0	0
$8$	1.50793	0.533135
$9$	1.84630	0.615433
$10$	−0.392048	−0.123977
$11$	0.255115	0.0769201	0.0384601	−	0.999260i	$-0.487755\pi$
$11$	0.255115	0.0769201	0.0384601	+	0.999260i	$0.487755\pi$
$12$	4.06450	1.17332
$13$	1.33836	0.371195	0.185598	−	0.982626i	$-0.440578\pi$
$13$	1.33836	0.371195	0.185598	+	0.982626i	$0.440578\pi$
$14$	0	0
$15$	−2.20143	−0.568407
$16$	3.10141	0.775353
$17$	−0.276162	−0.0669791	−0.0334896	−	0.999439i	$-0.510662\pi$
$17$	−0.276162	−0.0669791	−0.0334896	+	0.999439i	$0.510662\pi$
$18$	−0.723838	−0.170610
$19$	5.26593	1.20809	0.604043	−	0.796951i	$-0.293555\pi$
$19$	5.26593	1.20809	0.604043	+	0.796951i	$0.293555\pi$
$20$	−1.84630	−0.412845
$21$	0	0
$22$	−0.100017	−0.0213238
$23$	0.173848	0.0362499	0.0181249	−	0.999836i	$-0.494230\pi$
$23$	0.173848	0.0362499	0.0181249	+	0.999836i	$0.494230\pi$
$24$	−3.31961	−0.677613
$25$	1.00000	0.200000
$26$	−0.524703	−0.102903
$27$	2.53979	0.488784
$28$	0	0
$29$	−10.2196	−1.89773	−0.948866	−	0.315680i	$-0.897767\pi$
$29$	−10.2196	−1.89773	−0.948866	+	0.315680i	$0.897767\pi$
$30$	0.863067	0.157574
$31$	−2.40286	−0.431567	−0.215783	−	0.976441i	$-0.569230\pi$
$31$	−2.40286	−0.431567	−0.215783	+	0.976441i	$0.569230\pi$
$32$	−4.23177	−0.748079
$33$	−0.561619	−0.0977653
$34$	0.108269	0.0185680
$35$	0	0
$36$	−3.40882	−0.568136
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−2.06450	−0.334906
$39$	−2.94632	−0.471788
$40$	1.50793	0.238425
$41$	−1.01587	−0.158652	−0.0793260	−	0.996849i	$-0.525277\pi$
$41$	−1.01587	−0.158652	−0.0793260	+	0.996849i	$0.525277\pi$
$42$	0	0
$43$	−6.34995	−0.968359	−0.484179	−	0.874969i	$-0.660882\pi$
$43$	−6.34995	−0.968359	−0.484179	+	0.874969i	$0.660882\pi$
$44$	−0.471019	−0.0710088
$45$	1.84630	0.275230
$46$	−0.0681570	−0.0100492
$47$	1.32389	0.193109	0.0965547	−	0.995328i	$-0.469218\pi$
$47$	1.32389	0.193109	0.0965547	+	0.995328i	$0.469218\pi$
$48$	−6.82755	−0.985472
$49$	0	0
$50$	−0.392048	−0.0554440
$51$	0.607952	0.0851303
$52$	−2.47102	−0.342669
$53$	−5.58325	−0.766918	−0.383459	−	0.923558i	$-0.625267\pi$
$53$	−5.58325	−0.766918	−0.383459	+	0.923558i	$0.625267\pi$
$54$	−0.995722	−0.135501
$55$	0.255115	0.0343997
$56$	0	0
$57$	−11.5926	−1.53547
$58$	4.00657	0.526089
$59$	6.83412	0.889727	0.444863	−	0.895598i	$-0.353252\pi$
$59$	6.83412	0.889727	0.444863	+	0.895598i	$0.353252\pi$
$60$	4.06450	0.524724
$61$	−1.59348	−0.204024	−0.102012	−	0.994783i	$-0.532528\pi$
$61$	−1.59348	−0.204024	−0.102012	+	0.994783i	$0.532528\pi$
$62$	0.942038	0.119639
$63$	0	0
$64$	−4.54377	−0.567971
$65$	1.33836	0.166004
$66$	0.220182	0.0271025
$67$	−4.43838	−0.542235	−0.271117	−	0.962546i	$-0.587393\pi$
$67$	−4.43838	−0.542235	−0.271117	+	0.962546i	$0.587393\pi$
$68$	0.509878	0.0618317
$69$	−0.382715	−0.0460735
$70$	0	0
$71$	1.45974	0.173240	0.0866198	−	0.996241i	$-0.472393\pi$
$71$	1.45974	0.173240	0.0866198	+	0.996241i	$0.472393\pi$
$72$	2.78410	0.328109
$73$	6.54377	0.765890	0.382945	−	0.923771i	$-0.374910\pi$
$73$	6.54377	0.765890	0.382945	+	0.923771i	$0.374910\pi$
$74$	0.392048	0.0455747
$75$	−2.20143	−0.254199
$76$	−9.72248	−1.11524
$77$	0	0
$78$	1.15510	0.130789
$79$	2.54403	0.286226	0.143113	−	0.989706i	$-0.454289\pi$
$79$	2.54403	0.286226	0.143113	+	0.989706i	$0.454289\pi$
$80$	3.10141	0.346749
$81$	−11.1301	−1.23668
$82$	0.398269	0.0439815
$83$	3.29433	0.361599	0.180800	−	0.983520i	$-0.442131\pi$
$83$	3.29433	0.361599	0.180800	+	0.983520i	$0.442131\pi$
$84$	0	0
$85$	−0.276162	−0.0299540
$86$	2.48949	0.268448
$87$	22.4977	2.41201
$88$	0.384697	0.0410088
$89$	10.1693	1.07794	0.538971	−	0.842324i	$-0.318813\pi$
$89$	10.1693	1.07794	0.538971	+	0.842324i	$0.318813\pi$
$90$	−0.723838	−0.0762992
$91$	0	0
$92$	−0.320976	−0.0334641
$93$	5.28973	0.548520
$94$	−0.519029	−0.0535338
$95$	5.26593	0.540273
$96$	9.31595	0.950806
$97$	9.02672	0.916525	0.458262	−	0.888817i	$-0.348472\pi$
$97$	9.02672	0.916525	0.458262	+	0.888817i	$0.348472\pi$
$98$	0	0
$99$	0.471019	0.0473392
$100$	−1.84630	−0.184630
$101$	−17.2785	−1.71927	−0.859635	−	0.510908i	$-0.829309\pi$
$101$	−17.2785	−1.71927	−0.859635	+	0.510908i	$0.829309\pi$
$102$	−0.238346	−0.0235998
$103$	−9.68318	−0.954112	−0.477056	−	0.878873i	$-0.658296\pi$
$103$	−9.68318	−0.954112	−0.477056	+	0.878873i	$0.658296\pi$
$104$	2.01816	0.197897
$105$	0	0
$106$	2.18890	0.212605
$107$	8.81199	0.851888	0.425944	−	0.904750i	$-0.359942\pi$
$107$	8.81199	0.851888	0.425944	+	0.904750i	$0.359942\pi$
$108$	−4.68922	−0.451220
$109$	15.4903	1.48370	0.741851	−	0.670565i	$-0.233948\pi$
$109$	15.4903	1.48370	0.741851	+	0.670565i	$0.233948\pi$
$110$	−0.100017	−0.00953629
$111$	2.20143	0.208951
$112$	0	0
$113$	−3.28698	−0.309213	−0.154606	−	0.987976i	$-0.549411\pi$
$113$	−3.28698	−0.309213	−0.154606	+	0.987976i	$0.549411\pi$
$114$	4.54485	0.425664
$115$	0.173848	0.0162114
$116$	18.8684	1.75189
$117$	2.47102	0.228446
$118$	−2.67931	−0.246650
$119$	0	0
$120$	−3.31961	−0.303038
$121$	−10.9349	−0.994083
$122$	0.624721	0.0565595
$123$	2.23636	0.201646
$124$	4.43640	0.398400
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.2254	−0.996093	−0.498046	−	0.867150i	$-0.665949\pi$
$127$	−11.2254	−0.996093	−0.498046	+	0.867150i	$0.665949\pi$
$128$	10.2449	0.905532
$129$	13.9790	1.23078
$130$	−0.524703	−0.0460195
$131$	5.53949	0.483988	0.241994	−	0.970278i	$-0.422199\pi$
$131$	5.53949	0.483988	0.241994	+	0.970278i	$0.422199\pi$
$132$	1.03692	0.0902519
$133$	0	0
$134$	1.74006	0.150318
$135$	2.53979	0.218591
$136$	−0.416434	−0.0357089
$137$	9.50942	0.812444	0.406222	−	0.913774i	$-0.366846\pi$
$137$	9.50942	0.812444	0.406222	+	0.913774i	$0.366846\pi$
$138$	0.150043	0.0127725
$139$	−22.7763	−1.93186	−0.965930	−	0.258802i	$-0.916672\pi$
$139$	−22.7763	−1.93186	−0.965930	+	0.258802i	$0.916672\pi$
$140$	0	0
$141$	−2.91446	−0.245441
$142$	−0.572289	−0.0480255
$143$	0.341437	0.0285524
$144$	5.72613	0.477178
$145$	−10.2196	−0.848691
$146$	−2.56547	−0.212320
$147$	0	0
$148$	1.84630	0.151765
$149$	20.7343	1.69862	0.849311	−	0.527892i	$-0.177018\pi$
$149$	20.7343	1.69862	0.849311	+	0.527892i	$0.177018\pi$
$150$	0.863067	0.0704691
$151$	−3.27811	−0.266768	−0.133384	−	0.991064i	$-0.542584\pi$
$151$	−3.27811	−0.266768	−0.133384	+	0.991064i	$0.542584\pi$
$152$	7.94067	0.644074
$153$	−0.509878	−0.0412212
$154$	0	0
$155$	−2.40286	−0.193003
$156$	5.43978	0.435531
$157$	2.95137	0.235545	0.117773	−	0.993041i	$-0.462425\pi$
$157$	2.95137	0.235545	0.117773	+	0.993041i	$0.462425\pi$
$158$	−0.997384	−0.0793476
$159$	12.2911	0.974750
$160$	−4.23177	−0.334551
$161$	0	0
$162$	4.36353	0.342831
$163$	14.2091	1.11294	0.556471	−	0.830867i	$-0.312155\pi$
$163$	14.2091	1.11294	0.556471	+	0.830867i	$0.312155\pi$
$164$	1.87560	0.146459
$165$	−0.561619	−0.0437220
$166$	−1.29153	−0.100243
$167$	11.3353	0.877151	0.438575	−	0.898694i	$-0.355483\pi$
$167$	11.3353	0.877151	0.438575	+	0.898694i	$0.355483\pi$
$168$	0	0
$169$	−11.2088	−0.862214
$170$	0.108269	0.00830384
$171$	9.72248	0.743496
$172$	11.7239	0.893940
$173$	−11.5860	−0.880871	−0.440435	−	0.897784i	$-0.645176\pi$
$173$	−11.5860	−0.880871	−0.440435	+	0.897784i	$0.645176\pi$
$174$	−8.82020	−0.668657
$175$	0	0
$176$	0.791218	0.0596403
$177$	−15.0448	−1.13084
$178$	−3.98685	−0.298827
$179$	−6.09804	−0.455789	−0.227894	−	0.973686i	$-0.573184\pi$
$179$	−6.09804	−0.455789	−0.227894	+	0.973686i	$0.573184\pi$
$180$	−3.40882	−0.254078
$181$	9.50793	0.706719	0.353360	−	0.935488i	$-0.385039\pi$
$181$	9.50793	0.706719	0.353360	+	0.935488i	$0.385039\pi$
$182$	0	0
$183$	3.50793	0.259314
$184$	0.262152	0.0193261
$185$	−1.00000	−0.0735215
$186$	−2.07383	−0.152061
$187$	−0.0704532	−0.00515205
$188$	−2.44430	−0.178269
$189$	0	0
$190$	−2.06450	−0.149774
$191$	2.87212	0.207819	0.103910	−	0.994587i	$-0.466865\pi$
$191$	2.87212	0.207819	0.103910	+	0.994587i	$0.466865\pi$
$192$	10.0028	0.721889
$193$	−8.82386	−0.635155	−0.317578	−	0.948232i	$-0.602869\pi$
$193$	−8.82386	−0.635155	−0.317578	+	0.948232i	$0.602869\pi$
$194$	−3.53891	−0.254079
$195$	−2.94632	−0.210990
$196$	0	0
$197$	16.6963	1.18956	0.594779	−	0.803889i	$-0.297239\pi$
$197$	16.6963	1.18956	0.594779	+	0.803889i	$0.297239\pi$
$198$	−0.184662	−0.0131234
$199$	0.961637	0.0681686	0.0340843	−	0.999419i	$-0.489149\pi$
$199$	0.961637	0.0681686	0.0340843	+	0.999419i	$0.489149\pi$
$200$	1.50793	0.106627
$201$	9.77079	0.689178
$202$	6.77399	0.476616
$203$	0	0
$204$	−1.12246	−0.0785879
$205$	−1.01587	−0.0709513
$206$	3.79627	0.264499
$207$	0.320976	0.0223094
$208$	4.15082	0.287808
$209$	1.34342	0.0929262
$210$	0	0
$211$	−15.7622	−1.08511	−0.542557	−	0.840019i	$-0.682544\pi$
$211$	−15.7622	−1.08511	−0.542557	+	0.840019i	$0.682544\pi$
$212$	10.3083	0.707979
$213$	−3.21352	−0.220187
$214$	−3.45473	−0.236160
$215$	−6.34995	−0.433063
$216$	3.82984	0.260588
$217$	0	0
$218$	−6.07294	−0.411312
$219$	−14.4057	−0.973444
$220$	−0.471019	−0.0317561
$221$	−0.369605	−0.0248623
$222$	−0.863067	−0.0579253
$223$	5.78744	0.387555	0.193778	−	0.981045i	$-0.437926\pi$
$223$	5.78744	0.387555	0.193778	+	0.981045i	$0.437926\pi$
$224$	0	0
$225$	1.84630	0.123087
$226$	1.28865	0.0857199
$227$	−11.0453	−0.733103	−0.366551	−	0.930398i	$-0.619462\pi$
$227$	−11.0453	−0.733103	−0.366551	+	0.930398i	$0.619462\pi$
$228$	21.4034	1.41747
$229$	9.62747	0.636201	0.318101	−	0.948057i	$-0.396955\pi$
$229$	9.62747	0.636201	0.318101	+	0.948057i	$0.396955\pi$
$230$	−0.0681570	−0.00449414
$231$	0	0
$232$	−15.4105	−1.01175
$233$	6.25508	0.409784	0.204892	−	0.978785i	$-0.434316\pi$
$233$	6.25508	0.409784	0.204892	+	0.978785i	$0.434316\pi$
$234$	−0.968759	−0.0633297
$235$	1.32389	0.0863611
$236$	−12.6178	−0.821351
$237$	−5.60052	−0.363793
$238$	0	0
$239$	−5.17277	−0.334598	−0.167299	−	0.985906i	$-0.553505\pi$
$239$	−5.17277	−0.334598	−0.167299	+	0.985906i	$0.553505\pi$
$240$	−6.82755	−0.440716
$241$	22.8866	1.47425	0.737126	−	0.675755i	$-0.236182\pi$
$241$	22.8866	1.47425	0.737126	+	0.675755i	$0.236182\pi$
$242$	4.28701	0.275580
$243$	16.8827	1.08303
$244$	2.94204	0.188345
$245$	0	0
$246$	−0.876763	−0.0559004
$247$	7.04773	0.448436
$248$	−3.62336	−0.230083
$249$	−7.25223	−0.459592
$250$	−0.392048	−0.0247953
$251$	27.9048	1.76134	0.880669	−	0.473732i	$-0.157094\pi$
$251$	27.9048	1.76134	0.880669	+	0.473732i	$0.157094\pi$
$252$	0	0
$253$	0.0443514	0.00278835
$254$	4.40090	0.276137
$255$	0.607952	0.0380714
$256$	5.07103	0.316940
$257$	−5.66014	−0.353070	−0.176535	−	0.984294i	$-0.556489\pi$
$257$	−5.66014	−0.353070	−0.176535	+	0.984294i	$0.556489\pi$
$258$	−5.48044	−0.341197
$259$	0	0
$260$	−2.47102	−0.153246
$261$	−18.8684	−1.16793
$262$	−2.17175	−0.134171
$263$	−5.88897	−0.363130	−0.181565	−	0.983379i	$-0.558116\pi$
$263$	−5.88897	−0.363130	−0.181565	+	0.983379i	$0.558116\pi$
$264$	−0.846884	−0.0521221
$265$	−5.58325	−0.342976
$266$	0	0
$267$	−22.3870	−1.37006
$268$	8.19458	0.500563
$269$	−4.78959	−0.292026	−0.146013	−	0.989283i	$-0.546644\pi$
$269$	−4.78959	−0.292026	−0.146013	+	0.989283i	$0.546644\pi$
$270$	−0.995722	−0.0605977
$271$	−12.3012	−0.747246	−0.373623	−	0.927581i	$-0.621885\pi$
$271$	−12.3012	−0.747246	−0.373623	+	0.927581i	$0.621885\pi$
$272$	−0.856493	−0.0519325
$273$	0	0
$274$	−3.72815	−0.225226
$275$	0.255115	0.0153840
$276$	0.706607	0.0425327
$277$	10.3882	0.624166	0.312083	−	0.950055i	$-0.398973\pi$
$277$	10.3882	0.624166	0.312083	+	0.950055i	$0.398973\pi$
$278$	8.92941	0.535550
$279$	−4.43640	−0.265600
$280$	0	0
$281$	−1.14779	−0.0684711	−0.0342356	−	0.999414i	$-0.510900\pi$
$281$	−1.14779	−0.0684711	−0.0342356	+	0.999414i	$0.510900\pi$
$282$	1.14261	0.0680413
$283$	−25.4270	−1.51148	−0.755738	−	0.654874i	$-0.772722\pi$
$283$	−25.4270	−1.51148	−0.755738	+	0.654874i	$0.772722\pi$
$284$	−2.69512	−0.159926
$285$	−11.5926	−0.686685
$286$	−0.133860	−0.00791529
$287$	0	0
$288$	−7.81311	−0.460392
$289$	−16.9237	−0.995514
$290$	4.00657	0.235274
$291$	−19.8717	−1.16490
$292$	−12.0817	−0.707031
$293$	−32.2921	−1.88653	−0.943264	−	0.332044i	$-0.892262\pi$
$293$	−32.2921	−1.88653	−0.943264	+	0.332044i	$0.892262\pi$
$294$	0	0
$295$	6.83412	0.397898
$296$	−1.50793	−0.0876469
$297$	0.647940	0.0375973
$298$	−8.12886	−0.470892
$299$	0.232672	0.0134558
$300$	4.06450	0.234664
$301$	0	0
$302$	1.28518	0.0739535
$303$	38.0373	2.18519
$304$	16.3318	0.936694
$305$	−1.59348	−0.0912424
$306$	0.199897	0.0114273
$307$	18.3806	1.04904	0.524519	−	0.851399i	$-0.324245\pi$
$307$	18.3806	1.04904	0.524519	+	0.851399i	$0.324245\pi$
$308$	0	0
$309$	21.3168	1.21267
$310$	0.942038	0.0535041
$311$	−24.0525	−1.36389	−0.681946	−	0.731403i	$-0.738866\pi$
$311$	−24.0525	−1.36389	−0.681946	+	0.731403i	$0.738866\pi$
$312$	−4.44285	−0.251527
$313$	−13.4720	−0.761480	−0.380740	−	0.924682i	$-0.624331\pi$
$313$	−13.4720	−0.761480	−0.380740	+	0.924682i	$0.624331\pi$
$314$	−1.15708	−0.0652978
$315$	0	0
$316$	−4.69705	−0.264229
$317$	26.7938	1.50489	0.752445	−	0.658655i	$-0.228874\pi$
$317$	26.7938	1.50489	0.752445	+	0.658655i	$0.228874\pi$
$318$	−4.81872	−0.270220
$319$	−2.60717	−0.145974
$320$	−4.54377	−0.254004
$321$	−19.3990	−1.08275
$322$	0	0
$323$	−1.45425	−0.0809166
$324$	20.5494	1.14164
$325$	1.33836	0.0742391
$326$	−5.57064	−0.308529
$327$	−34.1008	−1.88578
$328$	−1.53186	−0.0845830
$329$	0	0
$330$	0.220182	0.0121206
$331$	5.60562	0.308113	0.154056	−	0.988062i	$-0.450766\pi$
$331$	5.60562	0.308113	0.154056	+	0.988062i	$0.450766\pi$
$332$	−6.08231	−0.333810
$333$	−1.84630	−0.101177
$334$	−4.44398	−0.243164
$335$	−4.43838	−0.242495
$336$	0	0
$337$	12.7245	0.693145	0.346573	−	0.938023i	$-0.387345\pi$
$337$	12.7245	0.693145	0.346573	+	0.938023i	$0.387345\pi$
$338$	4.39438	0.239023
$339$	7.23605	0.393008
$340$	0.509878	0.0276520
$341$	−0.613007	−0.0331962
$342$	−3.81168	−0.206112
$343$	0	0
$344$	−9.57531	−0.516266
$345$	−0.382715	−0.0206047
$346$	4.54229	0.244195
$347$	17.6319	0.946531	0.473265	−	0.880920i	$-0.343075\pi$
$347$	17.6319	0.946531	0.473265	+	0.880920i	$0.343075\pi$
$348$	−41.5375	−2.22665
$349$	13.3124	0.712597	0.356299	−	0.934372i	$-0.384039\pi$
$349$	13.3124	0.712597	0.356299	+	0.934372i	$0.384039\pi$
$350$	0	0
$351$	3.39917	0.181434
$352$	−1.07959	−0.0575423
$353$	9.42247	0.501508	0.250754	−	0.968051i	$-0.419322\pi$
$353$	9.42247	0.501508	0.250754	+	0.968051i	$0.419322\pi$
$354$	5.89831	0.313491
$355$	1.45974	0.0774751
$356$	−18.7755	−0.995102
$357$	0	0
$358$	2.39072	0.126354
$359$	−25.1416	−1.32692	−0.663462	−	0.748210i	$-0.730914\pi$
$359$	−25.1416	−1.32692	−0.663462	+	0.748210i	$0.730914\pi$
$360$	2.78410	0.146735
$361$	8.73001	0.459474
$362$	−3.72757	−0.195917
$363$	24.0725	1.26348
$364$	0	0
$365$	6.54377	0.342516
$366$	−1.37528	−0.0718870
$367$	1.42744	0.0745120	0.0372560	−	0.999306i	$-0.488138\pi$
$367$	1.42744	0.0745120	0.0372560	+	0.999306i	$0.488138\pi$
$368$	0.539176	0.0281065
$369$	−1.87560	−0.0976396
$370$	0.392048	0.0203816
$371$	0	0
$372$	−9.76643	−0.506366
$373$	1.53186	0.0793169	0.0396584	−	0.999213i	$-0.487373\pi$
$373$	1.53186	0.0793169	0.0396584	+	0.999213i	$0.487373\pi$
$374$	0.0276210	0.00142825
$375$	−2.20143	−0.113681
$376$	1.99634	0.102953
$377$	−13.6775	−0.704429
$378$	0	0
$379$	−5.31424	−0.272974	−0.136487	−	0.990642i	$-0.543581\pi$
$379$	−5.31424	−0.272974	−0.136487	+	0.990642i	$0.543581\pi$
$380$	−9.72248	−0.498752
$381$	24.7119	1.26603
$382$	−1.12601	−0.0576116
$383$	4.68827	0.239560	0.119780	−	0.992800i	$-0.461781\pi$
$383$	4.68827	0.239560	0.119780	+	0.992800i	$0.461781\pi$
$384$	−22.5535	−1.15093
$385$	0	0
$386$	3.45938	0.176078
$387$	−11.7239	−0.595960
$388$	−16.6660	−0.846089
$389$	−9.65184	−0.489368	−0.244684	−	0.969603i	$-0.578684\pi$
$389$	−9.65184	−0.489368	−0.244684	+	0.969603i	$0.578684\pi$
$390$	1.15510	0.0584906
$391$	−0.0480103	−0.00242799
$392$	0	0
$393$	−12.1948	−0.615147
$394$	−6.54573	−0.329769
$395$	2.54403	0.128004
$396$	−0.869641	−0.0437011
$397$	5.67953	0.285048	0.142524	−	0.989791i	$-0.454478\pi$
$397$	5.67953	0.285048	0.142524	+	0.989791i	$0.454478\pi$
$398$	−0.377008	−0.0188977
$399$	0	0
$400$	3.10141	0.155071
$401$	−16.1379	−0.805887	−0.402943	−	0.915225i	$-0.632013\pi$
$401$	−16.1379	−0.805887	−0.402943	+	0.915225i	$0.632013\pi$
$402$	−3.83062	−0.191054
$403$	−3.21590	−0.160196
$404$	31.9012	1.58714
$405$	−11.1301	−0.553058
$406$	0	0
$407$	−0.255115	−0.0126456
$408$	0.916751	0.0453860
$409$	−23.6728	−1.17054	−0.585271	−	0.810838i	$-0.699012\pi$
$409$	−23.6728	−1.17054	−0.585271	+	0.810838i	$0.699012\pi$
$410$	0.398269	0.0196691
$411$	−20.9343	−1.03261
$412$	17.8780	0.880788
$413$	0	0
$414$	−0.125838	−0.00618460
$415$	3.29433	0.161712
$416$	−5.66365	−0.277683
$417$	50.1405	2.45539
$418$	−0.526685	−0.0257610
$419$	−16.6740	−0.814580	−0.407290	−	0.913299i	$-0.633526\pi$
$419$	−16.6740	−0.814580	−0.407290	+	0.913299i	$0.633526\pi$
$420$	0	0
$421$	−18.9406	−0.923106	−0.461553	−	0.887113i	$-0.652708\pi$
$421$	−18.9406	−0.923106	−0.461553	+	0.887113i	$0.652708\pi$
$422$	6.17954	0.300815
$423$	2.44430	0.118846
$424$	−8.41917	−0.408871
$425$	−0.276162	−0.0133958
$426$	1.25986	0.0610402
$427$	0	0
$428$	−16.2696	−0.786419
$429$	−0.751650	−0.0362900
$430$	2.48949	0.120054
$431$	−27.8389	−1.34095	−0.670477	−	0.741930i	$-0.733911\pi$
$431$	−27.8389	−1.34095	−0.670477	+	0.741930i	$0.733911\pi$
$432$	7.87695	0.378980
$433$	23.9871	1.15275	0.576374	−	0.817186i	$-0.304467\pi$
$433$	23.9871	1.15275	0.576374	+	0.817186i	$0.304467\pi$
$434$	0	0
$435$	22.4977	1.07868
$436$	−28.5997	−1.36968
$437$	0.915473	0.0437930
$438$	5.64771	0.269858
$439$	−11.4703	−0.547448	−0.273724	−	0.961808i	$-0.588256\pi$
$439$	−11.4703	−0.547448	−0.273724	+	0.961808i	$0.588256\pi$
$440$	0.384697	0.0183397
$441$	0	0
$442$	0.144903	0.00689234
$443$	−0.376519	−0.0178889	−0.00894447	−	0.999960i	$-0.502847\pi$
$443$	−0.376519	−0.0178889	−0.00894447	+	0.999960i	$0.502847\pi$
$444$	−4.06450	−0.192893
$445$	10.1693	0.482071
$446$	−2.26895	−0.107438
$447$	−45.6452	−2.15894
$448$	0	0
$449$	29.1023	1.37342	0.686712	−	0.726930i	$-0.259053\pi$
$449$	29.1023	1.37342	0.686712	+	0.726930i	$0.259053\pi$
$450$	−0.723838	−0.0341220
$451$	−0.259164	−0.0122035
$452$	6.06874	0.285449
$453$	7.21652	0.339062
$454$	4.33029	0.203231
$455$	0	0
$456$	−17.4808	−0.818616
$457$	22.5171	1.05331	0.526653	−	0.850080i	$-0.323447\pi$
$457$	22.5171	1.05331	0.526653	+	0.850080i	$0.323447\pi$
$458$	−3.77443	−0.176368
$459$	−0.701395	−0.0327383
$460$	−0.320976	−0.0149656
$461$	7.72947	0.359997	0.179999	−	0.983667i	$-0.442391\pi$
$461$	7.72947	0.359997	0.179999	+	0.983667i	$0.442391\pi$
$462$	0	0
$463$	−28.3234	−1.31630	−0.658149	−	0.752887i	$-0.728661\pi$
$463$	−28.3234	−1.31630	−0.658149	+	0.752887i	$0.728661\pi$
$464$	−31.6952	−1.47141
$465$	5.28973	0.245306
$466$	−2.45229	−0.113600
$467$	−14.2749	−0.660563	−0.330281	−	0.943883i	$-0.607144\pi$
$467$	−14.2749	−0.660563	−0.330281	+	0.943883i	$0.607144\pi$
$468$	−4.56224	−0.210890
$469$	0	0
$470$	−0.519029	−0.0239410
$471$	−6.49724	−0.299377
$472$	10.3054	0.474345
$473$	−1.61997	−0.0744863
$474$	2.19567	0.100851
$475$	5.26593	0.241617
$476$	0	0
$477$	−10.3083	−0.471986
$478$	2.02797	0.0927574
$479$	−0.352216	−0.0160932	−0.00804658	−	0.999968i	$-0.502561\pi$
$479$	−0.352216	−0.0160932	−0.00804658	+	0.999968i	$0.502561\pi$
$480$	9.31595	0.425213
$481$	−1.33836	−0.0610241
$482$	−8.97263	−0.408692
$483$	0	0
$484$	20.1891	0.917687
$485$	9.02672	0.409882
$486$	−6.61884	−0.300237
$487$	−12.1878	−0.552284	−0.276142	−	0.961117i	$-0.589056\pi$
$487$	−12.1878	−0.552284	−0.276142	+	0.961117i	$0.589056\pi$
$488$	−2.40286	−0.108772
$489$	−31.2803	−1.41454
$490$	0	0
$491$	23.0640	1.04086	0.520432	−	0.853903i	$-0.325771\pi$
$491$	23.0640	1.04086	0.520432	+	0.853903i	$0.325771\pi$
$492$	−4.12900	−0.186150
$493$	2.82227	0.127108
$494$	−2.76305	−0.124315
$495$	0.471019	0.0211707
$496$	−7.45227	−0.334617
$497$	0	0
$498$	2.84323	0.127408
$499$	−13.1905	−0.590486	−0.295243	−	0.955422i	$-0.595401\pi$
$499$	−13.1905	−0.590486	−0.295243	+	0.955422i	$0.595401\pi$
$500$	−1.84630	−0.0825690
$501$	−24.9538	−1.11486
$502$	−10.9400	−0.488278
$503$	7.93728	0.353906	0.176953	−	0.984219i	$-0.443376\pi$
$503$	7.93728	0.353906	0.176953	+	0.984219i	$0.443376\pi$
$504$	0	0
$505$	−17.2785	−0.768881
$506$	−0.0173879	−0.000772986 0
$507$	24.6754	1.09587
$508$	20.7254	0.919542
$509$	−22.1571	−0.982097	−0.491048	−	0.871132i	$-0.663386\pi$
$509$	−22.1571	−0.982097	−0.491048	+	0.871132i	$0.663386\pi$
$510$	−0.238346	−0.0105542
$511$	0	0
$512$	−22.4779	−0.993394
$513$	13.3744	0.590493
$514$	2.21905	0.0978780
$515$	−9.68318	−0.426692
$516$	−25.8094	−1.13619
$517$	0.337745	0.0148540
$518$	0	0
$519$	25.5059	1.11958
$520$	2.01816	0.0885024
$521$	−17.9314	−0.785587	−0.392793	−	0.919627i	$-0.628491\pi$
$521$	−17.9314	−0.785587	−0.392793	+	0.919627i	$0.628491\pi$
$522$	7.39733	0.323772
$523$	−28.4369	−1.24346	−0.621730	−	0.783232i	$-0.713570\pi$
$523$	−28.4369	−1.24346	−0.621730	+	0.783232i	$0.713570\pi$
$524$	−10.2276	−0.446793
$525$	0	0
$526$	2.30876	0.100667
$527$	0.663579	0.0289060
$528$	−1.74181	−0.0758026
$529$	−22.9698	−0.998686
$530$	2.18890	0.0950798
$531$	12.6178	0.547567
$532$	0	0
$533$	−1.35960	−0.0588909
$534$	8.77678	0.379808
$535$	8.81199	0.380976
$536$	−6.69279	−0.289084
$537$	13.4244	0.579306
$538$	1.87775	0.0809555
$539$	0	0
$540$	−4.68922	−0.201792
$541$	−24.1728	−1.03927	−0.519635	−	0.854389i	$-0.673932\pi$
$541$	−24.1728	−1.03927	−0.519635	+	0.854389i	$0.673932\pi$
$542$	4.82267	0.207151
$543$	−20.9311	−0.898238
$544$	1.16866	0.0501057
$545$	15.4903	0.663531
$546$	0	0
$547$	−32.7697	−1.40113	−0.700566	−	0.713588i	$-0.747069\pi$
$547$	−32.7697	−1.40113	−0.700566	+	0.713588i	$0.747069\pi$
$548$	−17.5572	−0.750007
$549$	−2.94204	−0.125563
$550$	−0.100017	−0.00426476
$551$	−53.8157	−2.29262
$552$	−0.577109	−0.0245634
$553$	0	0
$554$	−4.07267	−0.173031
$555$	2.20143	0.0934455
$556$	42.0518	1.78340
$557$	18.9243	0.801850	0.400925	−	0.916111i	$-0.368689\pi$
$557$	18.9243	0.801850	0.400925	+	0.916111i	$0.368689\pi$
$558$	1.73928	0.0736297
$559$	−8.49855	−0.359450
$560$	0	0
$561$	0.155098	0.00654823
$562$	0.449987	0.0189816
$563$	4.75796	0.200524	0.100262	−	0.994961i	$-0.468032\pi$
$563$	4.75796	0.200524	0.100262	+	0.994961i	$0.468032\pi$
$564$	5.38095	0.226579
$565$	−3.28698	−0.138284
$566$	9.96860	0.419012
$567$	0	0
$568$	2.20120	0.0923601
$569$	−28.3509	−1.18853	−0.594266	−	0.804268i	$-0.702557\pi$
$569$	−28.3509	−1.18853	−0.594266	+	0.804268i	$0.702557\pi$
$570$	4.54485	0.190363
$571$	−30.6804	−1.28393	−0.641967	−	0.766732i	$-0.721882\pi$
$571$	−30.6804	−1.28393	−0.641967	+	0.766732i	$0.721882\pi$
$572$	−0.630395	−0.0263581
$573$	−6.32277	−0.264138
$574$	0	0
$575$	0.173848	0.00724998
$576$	−8.38915	−0.349548
$577$	39.3644	1.63876	0.819380	−	0.573250i	$-0.194318\pi$
$577$	39.3644	1.63876	0.819380	+	0.573250i	$0.194318\pi$
$578$	6.63492	0.275976
$579$	19.4251	0.807280
$580$	18.8684	0.783469
$581$	0	0
$582$	7.79067	0.322933
$583$	−1.42437	−0.0589914
$584$	9.86757	0.408323
$585$	2.47102	0.102164
$586$	12.6601	0.522983
$587$	11.2905	0.466010	0.233005	−	0.972476i	$-0.425144\pi$
$587$	11.2905	0.466010	0.233005	+	0.972476i	$0.425144\pi$
$588$	0	0
$589$	−12.6533	−0.521370
$590$	−2.67931	−0.110305
$591$	−36.7556	−1.51193
$592$	−3.10141	−0.127467
$593$	0.844669	0.0346864	0.0173432	−	0.999850i	$-0.494479\pi$
$593$	0.844669	0.0346864	0.0173432	+	0.999850i	$0.494479\pi$
$594$	−0.254024	−0.0104227
$595$	0	0
$596$	−38.2818	−1.56808
$597$	−2.11698	−0.0866421
$598$	−0.0912188	−0.00373021
$599$	8.68728	0.354952	0.177476	−	0.984125i	$-0.443207\pi$
$599$	8.68728	0.354952	0.177476	+	0.984125i	$0.443207\pi$
$600$	−3.31961	−0.135523
$601$	21.9854	0.896802	0.448401	−	0.893833i	$-0.351994\pi$
$601$	21.9854	0.896802	0.448401	+	0.893833i	$0.351994\pi$
$602$	0	0
$603$	−8.19458	−0.333709
$604$	6.05236	0.246267
$605$	−10.9349	−0.444568
$606$	−14.9125	−0.605777
$607$	−9.03808	−0.366844	−0.183422	−	0.983034i	$-0.558718\pi$
$607$	−9.03808	−0.366844	−0.183422	+	0.983034i	$0.558718\pi$
$608$	−22.2842	−0.903744
$609$	0	0
$610$	0.624721	0.0252942
$611$	1.77185	0.0716813
$612$	0.941386	0.0380533
$613$	−33.7654	−1.36377	−0.681885	−	0.731459i	$-0.738840\pi$
$613$	−33.7654	−1.36377	−0.681885	+	0.731459i	$0.738840\pi$
$614$	−7.20609	−0.290814
$615$	2.23636	0.0901789
$616$	0	0
$617$	3.04136	0.122440	0.0612202	−	0.998124i	$-0.480501\pi$
$617$	3.04136	0.122440	0.0612202	+	0.998124i	$0.480501\pi$
$618$	−8.35723	−0.336177
$619$	8.01154	0.322011	0.161006	−	0.986954i	$-0.448526\pi$
$619$	8.01154	0.322011	0.161006	+	0.986954i	$0.448526\pi$
$620$	4.43640	0.178170
$621$	0.441539	0.0177184
$622$	9.42974	0.378098
$623$	0	0
$624$	−9.13774	−0.365802
$625$	1.00000	0.0400000
$626$	5.28165	0.211097
$627$	−2.95744	−0.118109
$628$	−5.44911	−0.217443
$629$	0.276162	0.0110113
$630$	0	0
$631$	−47.3472	−1.88486	−0.942432	−	0.334398i	$-0.891467\pi$
$631$	−47.3472	−1.88486	−0.942432	+	0.334398i	$0.891467\pi$
$632$	3.83624	0.152597
$633$	34.6994	1.37918
$634$	−10.5045	−0.417186
$635$	−11.2254	−0.445466
$636$	−22.6931	−0.899840
$637$	0	0
$638$	1.02214	0.0404668
$639$	2.69512	0.106617
$640$	10.2449	0.404966
$641$	−29.2604	−1.15572	−0.577859	−	0.816137i	$-0.696112\pi$
$641$	−29.2604	−1.15572	−0.577859	+	0.816137i	$0.696112\pi$
$642$	7.60534	0.300159
$643$	−18.7096	−0.737835	−0.368917	−	0.929462i	$-0.620271\pi$
$643$	−18.7096	−0.737835	−0.368917	+	0.929462i	$0.620271\pi$
$644$	0	0
$645$	13.9790	0.550422
$646$	0.570136	0.0224317
$647$	−19.4974	−0.766523	−0.383262	−	0.923640i	$-0.625199\pi$
$647$	−19.4974	−0.766523	−0.383262	+	0.923640i	$0.625199\pi$
$648$	−16.7834	−0.659315
$649$	1.74349	0.0684379
$650$	−0.524703	−0.0205806
$651$	0	0
$652$	−26.2342	−1.02741
$653$	−4.89422	−0.191526	−0.0957629	−	0.995404i	$-0.530529\pi$
$653$	−4.89422	−0.191526	−0.0957629	+	0.995404i	$0.530529\pi$
$654$	13.3692	0.522776
$655$	5.53949	0.216446
$656$	−3.15063	−0.123011
$657$	12.0817	0.471354
$658$	0	0
$659$	15.8239	0.616413	0.308207	−	0.951319i	$-0.400271\pi$
$659$	15.8239	0.616413	0.308207	+	0.951319i	$0.400271\pi$
$660$	1.03692	0.0403619
$661$	45.7332	1.77882	0.889408	−	0.457115i	$-0.151117\pi$
$661$	45.7332	1.77882	0.889408	+	0.457115i	$0.151117\pi$
$662$	−2.19767	−0.0854150
$663$	0.813661	0.0316000
$664$	4.96763	0.192781
$665$	0	0
$666$	0.723838	0.0280481
$667$	−1.77666	−0.0687926
$668$	−20.9283	−0.809741
$669$	−12.7406	−0.492582
$670$	1.74006	0.0672244
$671$	−0.406521	−0.0156936
$672$	0	0
$673$	−12.5412	−0.483426	−0.241713	−	0.970348i	$-0.577709\pi$
$673$	−12.5412	−0.483426	−0.241713	+	0.970348i	$0.577709\pi$
$674$	−4.98860	−0.192154
$675$	2.53979	0.0977567
$676$	20.6948	0.795952
$677$	20.0036	0.768799	0.384400	−	0.923167i	$-0.374408\pi$
$677$	20.0036	0.768799	0.384400	+	0.923167i	$0.374408\pi$
$678$	−2.83688	−0.108950
$679$	0	0
$680$	−0.416434	−0.0159695
$681$	24.3155	0.931771
$682$	0.240328	0.00920264
$683$	−20.3062	−0.776997	−0.388498	−	0.921449i	$-0.627006\pi$
$683$	−20.3062	−0.776997	−0.388498	+	0.921449i	$0.627006\pi$
$684$	−17.9506	−0.686358
$685$	9.50942	0.363336
$686$	0	0
$687$	−21.1942	−0.808610
$688$	−19.6938	−0.750820
$689$	−7.47241	−0.284676
$690$	0.150043	0.00571203
$691$	−23.1754	−0.881636	−0.440818	−	0.897597i	$-0.645312\pi$
$691$	−23.1754	−0.881636	−0.440818	+	0.897597i	$0.645312\pi$
$692$	21.3913	0.813175
$693$	0	0
$694$	−6.91256	−0.262397
$695$	−22.7763	−0.863954
$696$	33.9251	1.28593
$697$	0.280544	0.0106264
$698$	−5.21911	−0.197546
$699$	−13.7701	−0.520834
$700$	0	0
$701$	−44.0298	−1.66298	−0.831492	−	0.555537i	$-0.812513\pi$
$701$	−44.0298	−1.66298	−0.831492	+	0.555537i	$0.812513\pi$
$702$	−1.33264	−0.0502972
$703$	−5.26593	−0.198608
$704$	−1.15918	−0.0436884
$705$	−2.91446	−0.109765
$706$	−3.69406	−0.139028
$707$	0	0
$708$	27.7773	1.04393
$709$	−14.8918	−0.559274	−0.279637	−	0.960106i	$-0.590214\pi$
$709$	−14.8918	−0.559274	−0.279637	+	0.960106i	$0.590214\pi$
$710$	−0.572289	−0.0214776
$711$	4.69705	0.176153
$712$	15.3346	0.574689
$713$	−0.417734	−0.0156443
$714$	0	0
$715$	0.341437	0.0127690
$716$	11.2588	0.420761
$717$	11.3875	0.425274
$718$	9.85673	0.367850
$719$	7.23956	0.269990	0.134995	−	0.990846i	$-0.456898\pi$
$719$	7.23956	0.269990	0.134995	+	0.990846i	$0.456898\pi$
$720$	5.72613	0.213400
$721$	0	0
$722$	−3.42258	−0.127375
$723$	−50.3832	−1.87377
$724$	−17.5545	−0.652407
$725$	−10.2196	−0.379546
$726$	−9.43757	−0.350261
$727$	27.9049	1.03494	0.517469	−	0.855702i	$-0.326874\pi$
$727$	27.9049	1.03494	0.517469	+	0.855702i	$0.326874\pi$
$728$	0	0
$729$	−3.77589	−0.139848
$730$	−2.56547	−0.0949524
$731$	1.75362	0.0648599
$732$	−6.47669	−0.239385
$733$	−15.7218	−0.580698	−0.290349	−	0.956921i	$-0.593771\pi$
$733$	−15.7218	−0.580698	−0.290349	+	0.956921i	$0.593771\pi$
$734$	−0.559627	−0.0206562
$735$	0	0
$736$	−0.735687	−0.0271178
$737$	−1.13230	−0.0417088
$738$	0.735324	0.0270677
$739$	−5.48469	−0.201758	−0.100879	−	0.994899i	$-0.532165\pi$
$739$	−5.48469	−0.201758	−0.100879	+	0.994899i	$0.532165\pi$
$740$	1.84630	0.0678713
$741$	−15.5151	−0.569961
$742$	0	0
$743$	40.8565	1.49888	0.749441	−	0.662072i	$-0.230323\pi$
$743$	40.8565	1.49888	0.749441	+	0.662072i	$0.230323\pi$
$744$	7.97657	0.292435
$745$	20.7343	0.759647
$746$	−0.600564	−0.0219882
$747$	6.08231	0.222540
$748$	0.130078	0.00475611
$749$	0	0
$750$	0.863067	0.0315148
$751$	−2.82502	−0.103087	−0.0515433	−	0.998671i	$-0.516414\pi$
$751$	−2.82502	−0.103087	−0.0515433	+	0.998671i	$0.516414\pi$
$752$	4.10593	0.149728
$753$	−61.4306	−2.23865
$754$	5.36225	0.195282
$755$	−3.27811	−0.119302
$756$	0	0
$757$	7.91528	0.287686	0.143843	−	0.989601i	$-0.454054\pi$
$757$	7.91528	0.287686	0.143843	+	0.989601i	$0.454054\pi$
$758$	2.08344	0.0756739
$759$	−0.0976365	−0.00354398
$760$	7.94067	0.288039
$761$	6.97754	0.252936	0.126468	−	0.991971i	$-0.459636\pi$
$761$	6.97754	0.252936	0.126468	+	0.991971i	$0.459636\pi$
$762$	−9.68827	−0.350969
$763$	0	0
$764$	−5.30279	−0.191848
$765$	−0.509878	−0.0184347
$766$	−1.83803	−0.0664107
$767$	9.14654	0.330262
$768$	−11.1635	−0.402829
$769$	26.3394	0.949824	0.474912	−	0.880033i	$-0.342480\pi$
$769$	26.3394	0.949824	0.474912	+	0.880033i	$0.342480\pi$
$770$	0	0
$771$	12.4604	0.448751
$772$	16.2915	0.586343
$773$	−28.5637	−1.02736	−0.513682	−	0.857980i	$-0.671719\pi$
$773$	−28.5637	−1.02736	−0.513682	+	0.857980i	$0.671719\pi$
$774$	4.59634	0.165212
$775$	−2.40286	−0.0863133
$776$	13.6117	0.488632
$777$	0	0
$778$	3.78399	0.135663
$779$	−5.34949	−0.191665
$780$	5.43978	0.194775
$781$	0.372403	0.0133256
$782$	0.0188224	0.000673087 0
$783$	−25.9557	−0.927580
$784$	0	0
$785$	2.95137	0.105339
$786$	4.78095	0.170531
$787$	−40.6974	−1.45071	−0.725353	−	0.688377i	$-0.758324\pi$
$787$	−40.6974	−1.45071	−0.725353	+	0.688377i	$0.758324\pi$
$788$	−30.8263	−1.09814
$789$	12.9642	0.461537
$790$	−0.997384	−0.0354853
$791$	0	0
$792$	0.710265	0.0252382
$793$	−2.13265	−0.0757328
$794$	−2.22665	−0.0790209
$795$	12.2911	0.435921
$796$	−1.77547	−0.0629298
$797$	−52.7815	−1.86962	−0.934809	−	0.355152i	$-0.884429\pi$
$797$	−52.7815	−1.86962	−0.934809	+	0.355152i	$0.884429\pi$
$798$	0	0
$799$	−0.365609	−0.0129343
$800$	−4.23177	−0.149616
$801$	18.7755	0.663401
$802$	6.32682	0.223408
$803$	1.66942	0.0589124
$804$	−18.0398	−0.636214
$805$	0	0
$806$	1.26079	0.0444094
$807$	10.5439	0.371165
$808$	−26.0548	−0.916604
$809$	−38.0770	−1.33872	−0.669358	−	0.742940i	$-0.733431\pi$
$809$	−38.0770	−1.33872	−0.669358	+	0.742940i	$0.733431\pi$
$810$	4.36353	0.153319
$811$	−44.2388	−1.55343	−0.776717	−	0.629850i	$-0.783117\pi$
$811$	−44.2388	−1.55343	−0.776717	+	0.629850i	$0.783117\pi$
$812$	0	0
$813$	27.0803	0.949747
$814$	0.100017	0.00350561
$815$	14.2091	0.497722
$816$	1.88551	0.0660060
$817$	−33.4384	−1.16986
$818$	9.28086	0.324498
$819$	0	0
$820$	1.87560	0.0654987
$821$	−45.0145	−1.57102	−0.785508	−	0.618851i	$-0.787598\pi$
$821$	−45.0145	−1.57102	−0.785508	+	0.618851i	$0.787598\pi$
$822$	8.20726	0.286261
$823$	−51.0857	−1.78074	−0.890368	−	0.455241i	$-0.849553\pi$
$823$	−51.0857	−1.78074	−0.890368	+	0.455241i	$0.849553\pi$
$824$	−14.6016	−0.508671
$825$	−0.561619	−0.0195531
$826$	0	0
$827$	−46.5393	−1.61833	−0.809165	−	0.587582i	$-0.800080\pi$
$827$	−46.5393	−1.61833	−0.809165	+	0.587582i	$0.800080\pi$
$828$	−0.592617	−0.0205949
$829$	25.1523	0.873576	0.436788	−	0.899565i	$-0.356116\pi$
$829$	25.1523	0.873576	0.436788	+	0.899565i	$0.356116\pi$
$830$	−1.29153	−0.0448298
$831$	−22.8689	−0.793313
$832$	−6.08122	−0.210828
$833$	0	0
$834$	−19.6575	−0.680683
$835$	11.3353	0.392274
$836$	−2.48035	−0.0857848
$837$	−6.10278	−0.210943
$838$	6.53702	0.225818
$839$	−13.8687	−0.478801	−0.239400	−	0.970921i	$-0.576951\pi$
$839$	−13.8687	−0.478801	−0.239400	+	0.970921i	$0.576951\pi$
$840$	0	0
$841$	75.4401	2.60138
$842$	7.42561	0.255903
$843$	2.52677	0.0870266
$844$	29.1017	1.00172
$845$	−11.2088	−0.385594
$846$	−0.958283	−0.0329464
$847$	0	0
$848$	−17.3160	−0.594632
$849$	55.9757	1.92108
$850$	0.108269	0.00371359
$851$	−0.173848	−0.00595945
$852$	5.93312	0.203265
$853$	37.9715	1.30012	0.650059	−	0.759884i	$-0.274744\pi$
$853$	37.9715	1.30012	0.650059	+	0.759884i	$0.274744\pi$
$854$	0	0
$855$	9.72248	0.332502
$856$	13.2879	0.454171
$857$	20.7367	0.708351	0.354175	−	0.935179i	$-0.384762\pi$
$857$	20.7367	0.708351	0.354175	+	0.935179i	$0.384762\pi$
$858$	0.294683	0.0100603
$859$	44.2668	1.51036	0.755182	−	0.655516i	$-0.227549\pi$
$859$	44.2668	1.51036	0.755182	+	0.655516i	$0.227549\pi$
$860$	11.7239	0.399782
$861$	0	0
$862$	10.9142	0.371739
$863$	−25.2651	−0.860035	−0.430018	−	0.902821i	$-0.641493\pi$
$863$	−25.2651	−0.860035	−0.430018	+	0.902821i	$0.641493\pi$
$864$	−10.7478	−0.365649
$865$	−11.5860	−0.393937
$866$	−9.40411	−0.319565
$867$	37.2564	1.26529
$868$	0	0
$869$	0.649022	0.0220166
$870$	−8.82020	−0.299033
$871$	−5.94017	−0.201275
$872$	23.3584	0.791014
$873$	16.6660	0.564059
$874$	−0.358910	−0.0121403
$875$	0	0
$876$	26.5971	0.898634
$877$	26.6365	0.899452	0.449726	−	0.893167i	$-0.351522\pi$
$877$	26.6365	0.899452	0.449726	+	0.893167i	$0.351522\pi$
$878$	4.49692	0.151764
$879$	71.0889	2.39777
$880$	0.791218	0.0266720
$881$	−17.8515	−0.601431	−0.300716	−	0.953714i	$-0.597226\pi$
$881$	−17.8515	−0.601431	−0.300716	+	0.953714i	$0.597226\pi$
$882$	0	0
$883$	30.9160	1.04041	0.520203	−	0.854043i	$-0.325856\pi$
$883$	30.9160	1.04041	0.520203	+	0.854043i	$0.325856\pi$
$884$	0.682402	0.0229517
$885$	−15.0448	−0.505727
$886$	0.147614	0.00495917
$887$	−12.8238	−0.430580	−0.215290	−	0.976550i	$-0.569070\pi$
$887$	−12.8238	−0.430580	−0.215290	+	0.976550i	$0.569070\pi$
$888$	3.31961	0.111399
$889$	0	0
$890$	−3.98685	−0.133640
$891$	−2.83945	−0.0951252
$892$	−10.6853	−0.357771
$893$	6.97152	0.233293
$894$	17.8951	0.598502
$895$	−6.09804	−0.203835
$896$	0	0
$897$	−0.512212	−0.0171023
$898$	−11.4095	−0.380740
$899$	24.5563	0.818998
$900$	−3.40882	−0.113627
$901$	1.54188	0.0513675
$902$	0.101605	0.00338306
$903$	0	0
$904$	−4.95654	−0.164852
$905$	9.50793	0.316054
$906$	−2.82922	−0.0939947
$907$	17.6839	0.587186	0.293593	−	0.955931i	$-0.405149\pi$
$907$	17.6839	0.587186	0.293593	+	0.955931i	$0.405149\pi$
$908$	20.3929	0.676763
$909$	−31.9012	−1.05810
$910$	0	0
$911$	17.3150	0.573670	0.286835	−	0.957980i	$-0.407397\pi$
$911$	17.3150	0.573670	0.286835	+	0.957980i	$0.407397\pi$
$912$	−35.9534	−1.19054
$913$	0.840433	0.0278143
$914$	−8.82779	−0.291997
$915$	3.50793	0.115969
$916$	−17.7752	−0.587309
$917$	0	0
$918$	0.274981	0.00907571
$919$	35.7265	1.17851	0.589254	−	0.807948i	$-0.299422\pi$
$919$	35.7265	1.17851	0.589254	+	0.807948i	$0.299422\pi$
$920$	0.262152	0.00864290
$921$	−40.4637	−1.33332
$922$	−3.03033	−0.0997985
$923$	1.95367	0.0643057
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	11.1041	0.364904
$927$	−17.8780	−0.587192
$928$	43.2470	1.41965
$929$	43.7620	1.43578	0.717892	−	0.696155i	$-0.245107\pi$
$929$	43.7620	1.43578	0.717892	+	0.696155i	$0.245107\pi$
$930$	−2.07383	−0.0680036
$931$	0	0
$932$	−11.5487	−0.378291
$933$	52.9499	1.73350
$934$	5.59644	0.183121
$935$	−0.0704532	−0.00230406
$936$	3.72613	0.121793
$937$	58.1375	1.89927	0.949635	−	0.313359i	$-0.101454\pi$
$937$	58.1375	1.89927	0.949635	+	0.313359i	$0.101454\pi$
$938$	0	0
$939$	29.6576	0.967838
$940$	−2.44430	−0.0797242
$941$	−36.8277	−1.20055	−0.600274	−	0.799795i	$-0.704942\pi$
$941$	−36.8277	−1.20055	−0.600274	+	0.799795i	$0.704942\pi$
$942$	2.54723	0.0829933
$943$	−0.176607	−0.00575112
$944$	21.1954	0.689853
$945$	0	0
$946$	0.635106	0.0206491
$947$	13.8629	0.450483	0.225241	−	0.974303i	$-0.427683\pi$
$947$	13.8629	0.450483	0.225241	+	0.974303i	$0.427683\pi$
$948$	10.3402	0.335835
$949$	8.75794	0.284295
$950$	−2.06450	−0.0669812
$951$	−58.9847	−1.91271
$952$	0	0
$953$	23.5558	0.763047	0.381524	−	0.924359i	$-0.375400\pi$
$953$	23.5558	0.763047	0.381524	+	0.924359i	$0.375400\pi$
$954$	4.04137	0.130844
$955$	2.87212	0.0929396
$956$	9.55047	0.308884
$957$	5.73952	0.185532
$958$	0.138086	0.00446135
$959$	0	0
$960$	10.0028	0.322839
$961$	−25.2263	−0.813750
$962$	0.524703	0.0169171
$963$	16.2696	0.524279
$964$	−42.2554	−1.36095
$965$	−8.82386	−0.284050
$966$	0	0
$967$	−33.4841	−1.07678	−0.538388	−	0.842697i	$-0.680967\pi$
$967$	−33.4841	−1.07678	−0.538388	+	0.842697i	$0.680967\pi$
$968$	−16.4891	−0.529981
$969$	3.20143	0.102845
$970$	−3.53891	−0.113628
$971$	−55.5264	−1.78193	−0.890963	−	0.454076i	$-0.849969\pi$
$971$	−55.5264	−1.78193	−0.890963	+	0.454076i	$0.849969\pi$
$972$	−31.1705	−0.999795
$973$	0	0
$974$	4.77822	0.153104
$975$	−2.94632	−0.0943576
$976$	−4.94204	−0.158191
$977$	5.68462	0.181867	0.0909335	−	0.995857i	$-0.471015\pi$
$977$	5.68462	0.181867	0.0909335	+	0.995857i	$0.471015\pi$
$978$	12.2634	0.392140
$979$	2.59434	0.0829155
$980$	0	0
$981$	28.5997	0.913118
$982$	−9.04219	−0.288548
$983$	−4.76817	−0.152081	−0.0760404	−	0.997105i	$-0.524228\pi$
$983$	−4.76817	−0.152081	−0.0760404	+	0.997105i	$0.524228\pi$
$984$	3.37229	0.107505
$985$	16.6963	0.531987
$986$	−1.10646	−0.0352370
$987$	0	0
$988$	−13.0122	−0.413974
$989$	−1.10393	−0.0351029
$990$	−0.184662	−0.00586895
$991$	0.665517	0.0211409	0.0105704	−	0.999944i	$-0.496635\pi$
$991$	0.665517	0.0211409	0.0105704	+	0.999944i	$0.496635\pi$
$992$	10.1684	0.322846
$993$	−12.3404	−0.391610
$994$	0	0
$995$	0.961637	0.0304859
$996$	13.3898	0.424272
$997$	−56.3881	−1.78583	−0.892915	−	0.450226i	$-0.851343\pi$
$997$	−56.3881	−1.78583	−0.892915	+	0.450226i	$0.851343\pi$
$998$	5.17130	0.163695
$999$	−2.53979	−0.0803556

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.i.1.3		5
7.6	odd	2		1295.2.a.f.1.3	✓	5
35.34	odd	2		6475.2.a.p.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1295.2.a.f.1.3	✓	5	7.6	odd	2
6475.2.a.p.1.3		5	35.34	odd	2
9065.2.a.i.1.3		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9065.a (trivial)

\(f(q)\)	\(=\)	\(q-0.392048 q^{2} -2.20143 q^{3} -1.84630 q^{4} +1.00000 q^{5} +0.863067 q^{6} +1.50793 q^{8} +1.84630 q^{9} -0.392048 q^{10} +0.255115 q^{11} +4.06450 q^{12} +1.33836 q^{13} -2.20143 q^{15} +3.10141 q^{16} -0.276162 q^{17} -0.723838 q^{18} +5.26593 q^{19} -1.84630 q^{20} -0.100017 q^{22} +0.173848 q^{23} -3.31961 q^{24} +1.00000 q^{25} -0.524703 q^{26} +2.53979 q^{27} -10.2196 q^{29} +0.863067 q^{30} -2.40286 q^{31} -4.23177 q^{32} -0.561619 q^{33} +0.108269 q^{34} -3.40882 q^{36} -1.00000 q^{37} -2.06450 q^{38} -2.94632 q^{39} +1.50793 q^{40} -1.01587 q^{41} -6.34995 q^{43} -0.471019 q^{44} +1.84630 q^{45} -0.0681570 q^{46} +1.32389 q^{47} -6.82755 q^{48} -0.392048 q^{50} +0.607952 q^{51} -2.47102 q^{52} -5.58325 q^{53} -0.995722 q^{54} +0.255115 q^{55} -11.5926 q^{57} +4.00657 q^{58} +6.83412 q^{59} +4.06450 q^{60} -1.59348 q^{61} +0.942038 q^{62} -4.54377 q^{64} +1.33836 q^{65} +0.220182 q^{66} -4.43838 q^{67} +0.509878 q^{68} -0.382715 q^{69} +1.45974 q^{71} +2.78410 q^{72} +6.54377 q^{73} +0.392048 q^{74} -2.20143 q^{75} -9.72248 q^{76} +1.15510 q^{78} +2.54403 q^{79} +3.10141 q^{80} -11.1301 q^{81} +0.398269 q^{82} +3.29433 q^{83} -0.276162 q^{85} +2.48949 q^{86} +22.4977 q^{87} +0.384697 q^{88} +10.1693 q^{89} -0.723838 q^{90} -0.320976 q^{92} +5.28973 q^{93} -0.519029 q^{94} +5.26593 q^{95} +9.31595 q^{96} +9.02672 q^{97} +0.471019 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 2 q^{4} + 5 q^{5} - 2 q^{6} - 2 q^{9} - 7 q^{11} + 4 q^{12} + 3 q^{13} - q^{15} - 4 q^{16} - 5 q^{17} + 2 q^{20} - 10 q^{22} + 4 q^{23} + 8 q^{24} + 5 q^{25} - q^{27} - 17 q^{29} - 2 q^{30}+ \cdots - 2 q^{99}+O(q^{100}) \) 5 * q - q^3 + 2 * q^4 + 5 * q^5 - 2 * q^6 - 2 * q^9 - 7 * q^11 + 4 * q^12 + 3 * q^13 - q^15 - 4 * q^16 - 5 * q^17 + 2 * q^20 - 10 * q^22 + 4 * q^23 + 8 * q^24 + 5 * q^25 - q^27 - 17 * q^29 - 2 * q^30 + 8 * q^31 - 10 * q^32 + 3 * q^33 + 24 * q^34 - 20 * q^36 - 5 * q^37 + 6 * q^38 - 13 * q^39 + 10 * q^41 - 6 * q^43 + 2 * q^44 - 2 * q^45 - 16 * q^46 - 11 * q^47 - 2 * q^48 + 5 * q^51 - 8 * q^52 + 14 * q^53 + 2 * q^54 - 7 * q^55 - 26 * q^57 + 18 * q^58 + 4 * q^60 + 4 * q^61 - 4 * q^62 - 16 * q^64 + 3 * q^65 + 12 * q^66 - 28 * q^67 - 12 * q^68 + 10 * q^69 + 10 * q^72 + 26 * q^73 - q^75 - 4 * q^76 - 12 * q^78 - 17 * q^79 - 4 * q^80 - 19 * q^81 - 4 * q^83 - 5 * q^85 - 16 * q^86 + 5 * q^87 + 6 * q^88 + 2 * q^89 - 18 * q^92 + 24 * q^93 - 22 * q^94 - 10 * q^96 + 35 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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